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Appl. Phys. Lett. 114, 093704 (2019); https://doi.org/10.1063/1.5082727 114, 093704
© 2019 Author(s).
Macrotextures-induced jumping relay of
condensate droplets
Cite as: Appl. Phys. Lett. 114, 093704 (2019); https://doi.org/10.1063/1.5082727
Submitted: 23 November 2018 . Accepted: 12 February 2019 . Published Online: 08 March 2019
Yaqi Cheng , Bingang Du, Kai Wang, Yansong Chen, Zhong Lan, Zuankai Wang , and Xuehu Ma
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Macrotextures-induced jumping relay of
condensate droplets
Cite as: Appl. Phys. Lett. 114, 093704 (2019); doi: 10.1063/1.5082727
Submitted: 23 November 2018 .Accepted: 12 February 2019 .
Published Online: 8 March 2019
Yaqi Cheng,
1,2
Bingang Du,
1
Kai Wang,
1
Yansong Chen,
1
Zhong Lan,
1
Zuankai Wang,
2,a)
and Xuehu Ma
1,a)
AFFILIATIONS
1
State Key Laboratory of Fine Chemicals, Liaoning Key Laboratory of Clean Utilization of Chemical Resources, Institute of Chemical
Engineering, Dalian University of Technology, Dalian 116024, China
2
Department of Mechanical Engineering, City University of Hong Kong, Hong Kong 999077, China
a)
Authors to whom correspondence should be addressed: zuanwang@cityu.edu.hk and xuehuma@dlut.edu.cn
ABSTRACT
Self-propelled droplet jumping plays a crucial role in numerous applications such as condensation heat transfer, self-cleaning, and water
harvesting. Compared to individual droplet jumping, the coalescence-induced droplet jumping in a domino manner has attracted more
attention due to its potential for the high performance of droplet mobility and heat transfer. However, there is an apparent gap in the current
literature regarding the demonstration of the advantage of this preferred droplet transport in a well-controlled way. In this study, we report
the attainment of droplet jumping relay by designing a nanosheet-covered superhydrophobic surface with V-shaped macrogrooves (Groove-
SHS). We find that the macrogroove arrays can significantly modify the droplet dynamics in the presence of a non-condensable gas (NCG)
by coupling rapid droplet growth and efficient droplet removal by jumping relay. The condensate droplets formed through the NCG diffu-
sion layer on top of the cones and between the grooves serve as more efficient conduits for heat transfer. The droplets with higher mobility
formed on the bottom of the grooves can undergo a series of coalescence which results in the preferred droplet jumping relay. Such a droplet
jumping relay can induce a considerable vibration for triggering the removal of droplets on top of the cones. The condensation performance
of the Groove-SHS is increased by 60% compared to that of the flat superhydrophobic surface due to the synergistic effect of rapid droplet
growth and efficient droplet removal facilitated by the integration of the droplet jumping relay. The mechanisms revealed in this work pave
the way for dropwise condensation enhancement.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5082727
Dropwise condensation has shown great potential for improving
the energy transfer efficiency for a wide range of industrial applica-
tions, such as power generation, water desalination, and thermal
management.
1–4
Typically, the self-propelled jumping of droplet with
a smaller size than the capillary length (2.7 mm) which is achieved
by the release of surface free energy during droplet coalescence on
superhydrophobic surfaces,
5–7
can significantly improve the heat
transfer performance of dropwise condensation.
8–13
Such jumping
droplet condensation can be further enhanced by the minimization of
surface adhesion,
14–16
enhancement of droplet jumping velocity,
17–19
and integration of rapid droplet nucleation and removal.
20–22
Most of
these studies focus on the structure optimization on a micro-/nano-
scale that is comparable with that of condensate droplets in the nucle-
ation and growth stages. However, for vapor condensation in the pres-
ence of non-condensable gas (NCG), the thermal resistance for the
condensation process is dominated by the vapor mass transfer in a dif-
fusion layer near the liquid-vapor interface.
23
To improve the heat transfer performance of vapor condensation
in the presence of NCG, enhancement of the droplet mobility, such as
droplet coalescence, droplet sliding or droplet jumping, is desired to
promote the disturbance of the diffusion layer.
24
In particular, an
advanced strategy is to activate successive droplet jumping with a
domino effect to maximize the mobility of droplets.
25–27
Nevertheless,
it is difficult for the final condensate droplets to depart from the hori-
zontally oriented surfaces due to the gravity effect and viscous energy
dissipation, which is not beneficial to the overall heat transfer enhance-
ment.
28
Although successive coalescence-induced droplet jumping
relay or sweeping has also been observed and reported for the verti-
cally oriented superhydrophobic surfaces, these phenomena are ran-
dom and not regulated.
29–32
To date, the rational structure design to
demonstrate the advantages of droplet jumping relay for condensation
enhancement is still lacking.
Recent studies show that macrotextures in a millimeter-scale can
remarkably modify the droplet growth rate and the spatial distribution
Appl. Phys. Lett. 114, 093704 (2019); doi: 10.1063/1.5082727 114, 093704-1
Published under license by AIP Publishing
Applied Physics Letters ARTICLE scitation.org/journal/apl
during condensation in the presence of NCG.
33,34
On macrotextures,
droplets can grow faster and larger on the top than on the bottom due
to higher diffusion flux of water vapor, which provides a potential
opportunity for condensation enhancement in the presence of NCG.
Here, we present a nanosheet-covered superhydrophobic surface
with V-shaped macrogrooves that facilitate a well-controlled droplet
jumping relay in condensation in the presence of NCG. To address the
limitation of vapor mass transfer, we design the depth of the grooves
to be comparable with the thickness of the diffusion layer on the milli-
meter scale. Such a structure design integrates the rapid droplet
growth on top of cones and the efficient droplet jumping relay, as well
as increased surface renewal (Fig. 1). Figure 1(b) shows a decreasing
distribution of NCG concentration (C
NCG
) in the diffusion layer
towards the top of the groove. Due to a smaller NCG concentration,
the droplets on top of the grooves can transfer heat and mass more
efficiently with a higher growth rate. Meanwhile, these “wetted” drop-
lets in the region of smaller NCG concentration prefer toget immersed
into the nanosheets, i.e., partially wetted or Wenzel modes, leading to
the increase of the liquid-solid fraction (f
sl
) and the surface adhesion
work (W
ad
). Interestingly, the small mobile droplets in the Cassie state
on the bottom of the grooves are more accessible to jump and to
trigger the preferred droplet jumping relay through successive coales-
cence, by which the removal of the rapid growing droplets on top of
the grooves is also promoted. The synergy between rapid droplet
growth and efficient droplet departure by droplet jumping relay on the
functionally partitioned surface is expected to enhance the heat trans-
fer performance of condensation in the presence of NCG.
To demonstrate the concept of condensation enhancement, we
fabricate a macrogrooved superhydrophobic surface (Groove-SHS)
and a flat superhydrophobic surface (Flat-SHS) for comparison. The
droplet dynamics and the condensation performance are characterized
in a custom-built condensation chamber. The detailed sample prepa-
ration, characterization, and the experimental setup are listed in the
supplementary material S2 and S3.
Figure 2 compares the distribution of condensate droplets on
the Flat-SHS and the Groove-SHS. The droplets on the Flat-SHS are
randomly distributed [Fig. 2(a)], which have been widely studied in
literature.
35–37
In contrast, the droplets on the Groove-SHS exhibit a
preferential spatial distribution [Fig. 2(b),alsoseethesupplementary
material S4 for the droplet distribution from a side view and details of
the droplet growth], due to the concentration gradient of water vapor
along the macrogrooves. First, the higher-concentration water vapor
on top of the cones leads to a faster droplet growth with a larger
vapor mass flux.
38
Second, the immersed wetting states of the conden-
sate droplets on top of cones are preferred under higher vapor satura-
tion, which can lead to the increase of surface adhesion for droplet
departure.
39
Meanwhile, the direct solid-liquid contact between the
droplets and nanosheets reduces the thermal resistance of the vapor
film under the droplets which can further improve the droplet growth.
For the droplets distributed in the grooves toward the bottom, the
growth rate decreases due to the increased concentration of NCG, but
the droplet mobility increases attributed to the suspended states. Such
an interesting droplet distribution with the evolution of wetting states
and mobility along the grooves makes it possible to activate a direc-
tional droplet jumping relay from the bottom to the top of the grooves.
During the condensation experiments, we find droplet jumping
relay shows two different modes, short-range jumping relay in the
grooves and long-range jumping relay on the cones. Figure 3 (multi-
media view) shows the short-range droplet jumping relay. The droplet
coalescence and the jumping relay are primarily investigated by the
trace line due to the high speed of droplet jumping. The seed droplet
(initial small droplet on the bottom of the groove) before jumping is
highlighted with dashed red circles at t¼0sinFig. 3(a). After the
droplet jumps off the wall at t¼0.008 s, it impacts another small drop-
let on the opposite surface in the groove, triggering a new coalescence
and jumping at t¼0.016 s. At t¼0.024 s, the jumping droplet driven
by successive coalescence climbs to the top of the cone. Further coales-
cencewithotherdropletsgrowingontopoftheconetriggersthefinal
departure of the condensate droplets, thus completing the droplet
jumping relay along the groove. Due to the well-controlled droplet
direction in the V-shaped grooves, the short-range jumping relay on
the Groove-SHS significantly improves the departure of condensate
droplets compared to the traditional droplet jumping on the Flat-SHS.
Figure 4 (multimedia view) illustrates the vibration-induced
long-range droplet jumping relay. From t¼0stot¼0.024 s, the coa-
lesced droplet exhibits a long-range jumping when two small droplets
merge on top of the groove. However, the droplet jumping relay is
interrupted when the small jumping droplet merges with a big droplet
FIG. 1. Schematic of the working principle of the droplet jumping relay. (a) Top view
and the overall schematic of the droplet jumping relay. Small droplets on the bottom
of the groove successively jump and trigger the jumping relay, promoting the droplet
departure on top of the cones. (b) A cross-sectional view and a detailed schematic
of the droplet jumping relay. The formation of the concentration gradient of NCG
along the groove leads to a change in the droplet wetting state and the growth rate.
Small droplets can easily jump and trigger the jumping relay, accelerating the
departure of top droplets with a high growth rate.
FIG. 2. Water droplet distribution on different surfaces during condensation. (a) The
random distribution of droplets on the Flat-SHS. (b) The preferential spatial distribu-
tion of droplets on the Groove-SHS. The size of droplets increases with the groove
height.
Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 093704 (2019); doi: 10.1063/1.5082727 114, 093704-2
Published under license by AIP Publishing
growing on the top of the cone at t¼0.032 s. This is due to the mis-
match in size of the two coalesced droplets with a size ratio of 0.14,
which is far smaller than the critical droplet size ratio for the
coalescence-induced droplet jumping of 0.56.
40
Although there is no
immediate coalescence-induced jumping, the impact of the jumping
droplet causes a violent vibration of the coalesced droplet from
t¼0.032 s to t¼0.040 s. This droplet vibration increases its possibility
to merge with neighboring droplets, promoting droplet departure with
asmallersize,asshownatt¼0.048 s. Note that the small droplets in
the grooves are not removed by the departure droplet on the top of the
cone, indicating that the droplet departure caused by the vibration and
the coalescence is in the form of jumping. To clarify the underlying
mechanism, we conduct both the energy analysis and the force balance
analysis in the supplementary material S5. Based on the experimental
measurement, the effective range for the droplet to coalesce with
neighboring droplets is increased from 437 lm (vibrational droplet
radius) to 479 lm, resulting in 10% increase in the range around
the droplet to coalesce with the neighboring droplets compared with
droplets without vibration. Therefore, the vibration of the large droplet
on the cones has a critical effect on triggering successive effective
FIG. 3. Short-range droplet jumping relay.
Optical snapshots (a) and a schematic dia-
gram (b). The droplet jumping occurring on
the bottom of the groove triggers a fast and
short-range jumping relay, which enhances
the droplet departure. Multimedia view:
https://doi.org/10.1063/1.5082727.1
FIG. 4. Long-range droplet jumping relay.
The droplet jumping occurs on the top
side of the groove jumps and impacts a
large droplet on its trajectory. The impact
induces considerable vibration of the
coalesced droplet and stimulates further
coalescence-induced jumping relay depar-
ture. The droplets highlighted with orange
circles indicate that the final departure of
the droplet is in the way of jumping rather
than gravity-driven sliding. Multimedia
view: https://doi.org/10.1063/1.5082727.2
Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 093704 (2019); doi: 10.1063/1.5082727 114, 093704-3
Published under license by AIP Publishing
droplet coalescence, and thus the long-range droplet jumping relay
departure.
To quantitatively illustrate the significance of droplet vibration
for further droplet coalescence and jumping, we develop a vibration
model to analyze the vibration process caused by the droplet impact.
In the model, the droplet vibration induced by the jumping relay is
seen as a liquid spring that resembles the damped harmonic oscilla-
tion. To simplify the model and intuitively illustrate the effect of drop-
let jumping relay on the vibration, we first analyze the characteristic
time of droplet coalescence s
c
and droplet incoming s
i
. The result
shows that the droplet vibration energy mainly comes from the initial
momentum of the seed droplet (initial jumping droplet). The droplet
vibration range can be expressed by the damped harmonic oscillation
equation x¼AectcosðwdtþuÞ,whereAis the vibration amplitude
(in this case, the vibration amplitude equals the radius of the coalesced
droplet R
c
), cis the damping factor, w
d
is the damped angular fre-
quency, and uis the phase angle. The time period of droplet vibration
is dependent on the coalesced droplet radius R
c
,theliquiddensityq,
and the surface tension r
lv
, that can be written as T¼pffiffiffiffiffiffi
qR3
c
2rlv
q.The
vibrating droplet can reach its displacement maxima at 2iþ1
ðÞ
T
4(i0)
and keeps damping in the vibration process. The vibration range, i.e.,
the maximum coalescence range with neighboring droplets, is
achieved at t¼T/4. The detailed description of the model is shown in
the supplementary material S6. Here, we define an enlargement ratio,
i.e., the ratio of the vibration range to the droplet radius [see in Fig.
5(a)], to describe the effect of droplet vibration on the following drop-
let coalescence. Figures 5(b) and 5(c) show the effect of the droplet
radius and the velocity of the seed droplet on the enlargement ratio of
the vibration range, respectively. The results show that the
enlargement ratio of the vibration range increases with the increase in
the radius of the target droplet, the droplet radius ratio, and the veloc-
ity of the seed droplet. The calculated enlargement ratio of 11%
shows good agreement with the experimental result of 10% [Fig.
5(c)]. In addition to the initial momentum of the seed droplet, the
influence of the propagation of capillary waves on the expansion of the
coalesced droplet is also analyzed (see details in the supplementary
material S7). Thus, the vibratory kinetic energy can be greatly utilized
in promoting the long-range droplet jumping relay, accordingly
enhancing the droplet removal for accelerating surface refreshment.
We further study the effect of structure geometry on the modes
of droplet jumping relay [Fig. 6(a)]. To simplify the analysis, we
assume that (a) the first seed droplet jumps perpendicular to the sur-
face of the groove and (b) the postdroplet jumping obeys the reflection
law. The first jumping height of the seed droplet h
1
can be expressed
as h1¼1
22htan a
2þA
tan a,wherehis the height of the first seed
droplet before jumping, ais the angle of the V-shaped groove, and A
is the width of the ridge and the valley of the grooves, as illustrated in
Fig. 1(b). The total height of the droplet after ntime jumping can be
obtained as hn¼2Pn1
i¼0htana
2þA
sin 2n1
ðÞ
a
2cos a
2
cos2a(h
n
<H). For the grooves
with H¼1.0 mm, a¼30,andA¼0.10 mm in this study, the height
of droplet jumping is calculated in Fig. 6(a). It can be seen that the
droplets conduct a continuous jumping relay when 0 mm <h<
0.41 mm. With the increase in h(0.41 mm <h<0.85 mm), single
droplet jumping relay occurs, which is consistent with the experimen-
tal observations of the short-range droplet jumping relay in Fig. 3.
With the further increase in h(0.85 mm <h<1.00 mm), the droplet
departure of the long-range jumping relay mode happens, which is
also consistent with the experimental observations in Fig. 4. To further
FIG. 5. (a) Schematic of the jumping
relay-induced vibration. (b) The influence
of the droplet radius on the enlargement
ratio of the vibration range of the coa-
lesced droplet (v
s
¼0.35 m/s). (c) The
influence of the seed droplet velocity on
the enlargement ratio of the vibration range
of the coalesced droplet (R
t
¼437 lm).
FIG. 6. Droplet jumping relay and conden-
sation performance. (a) The theoretical
criterion of droplet jumping relay modes.
The solid red line and the solid black line
represent the height of the first time and
the second time jumping, respectively.
Inset: Schematic of the droplet jumping
relay with different modes. (b) Cumulative
departure volume of the condensate on
the Groove-SHS and the Flat-SHS (T
s
¼2.6 C, T
air
¼32 C, and RH ¼83%).
Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 093704 (2019); doi: 10.1063/1.5082727 114, 093704-4
Published under license by AIP Publishing
evaluate the effect of droplet jumping relay on the condensation
performance, condensation experiments are conducted on the
Groove-SHS and the Flat-SHS. Figure 6(b) shows that the cumulative
departure volume of the Groove-SHS exceeds about 60% of that of the
Flat-SHS, which benefits from the integration of the fast droplet
growth and the efficient droplet departure.
In summary, we design and fabricate a nanosheet-covered super-
hydrophobic surface with macro-V-shaped groove arrays. A well-
controlled directional droplet jumping relay is demonstrated during
vapor condensation in the presence of NCG, which is due to the pref-
erential droplet spatial distribution with the evolution of wetting states
and mobility along the grooves. We further find that the jumping relay
of small droplets can lead to a violent vibration of the large target
droplet on the top of the cones. Quantitative analysis of the droplet
vibration suggests that the vibratory energy can enlarge the effective
coalescence range, leading to long-range jumping relay and efficient
removal of the large droplets growing on the cones. Such a synergistic
effect of rapid droplet growth and efficient droplet jumping relay on
the functionally partitioned surface provides a visible way for dropwise
condensation enhancement.
See supplementary material for more details about this work.
The authors are grateful for the financial support by the National
Natural Science Foundation of China (No. 51836002), the Science and
Technology Project in Dalian (No. 2015E11SF064) and Research
Grants Council of Hong Kong (Nos. C1018-17G, 11275216, and
11218417). The authors would like to thank Dr. Rongfu Wen from the
University of Colorado Boulder for the support in writing the
manuscript.
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Appl. Phys. Lett. 114, 093704 (2019); doi: 10.1063/1.5082727 114, 093704-5
Published under license by AIP Publishing
